Learning Objectives

By the end of this section, you will be able to:

- Determine the degree of polynomials
- Add and subtract polynomials
- Evaluate a polynomial function for a given value
- Add and subtract polynomial functions

Note

Before you get started, take this readiness quiz.

- Simplify: (3x^2+3x+1+8x^2+5x+5.)

If you missed this problem, review__[link]__. - Subtract: ((5n+8)−(2n−1).)

If you missed this problem, review__[link]__. - Evaluate: (4xy^2) when (x=−2x) and (y=5.).

If you missed this problem, review__[link]__.

## Determine the Degree of Polynomials

We have learned that a *term* is a constant or the product of a constant and one or more variables. A **monomial** is an algebraic expression with one term. When it is of the form (ax^m), where (a) is a constant and (m) is a whole number, it is called a monomial in one variable. Some examples of monomial in one variable are. Monomials can also have more than one variable such as and (−4a^2b^3c^2.)

Definition: MONOMIAL

A *monomial *is an algebraic expression with one term. A monomial in one variable is a term of the form (ax^m), where (a) is a constant and (m) is a whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a **polynomial**. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a **trinomial** has exactly three terms. There are no special names for polynomials with more than three terms.

Definition: POLYNOMIALS

*polynomial*—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.*monomial*—A polynomial with exactly one term is called a monomial.*binomial*—A polynomial with exactly two terms is called a binomial.*trinomial*—A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

Polynomial | (y+1) | (4a^2−7ab+2b^2) | (4x^4+x^3+8x^2−9x+1) | |
---|---|---|---|---|

Monomial | (14) | (8y^2) | (−9x^3y^5) | (−13a^3b^2c) |

Binomial | (a+7ba+7b) | (4x^2−y^2) | (y^2−16) | (3p^3q−9p^2q) |

Trinomial | (x^2−7x+12) | (9m^2+2mn−8n^2) | (6k^4−k^3+8k) | (z^4+3z^2−1) |

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words *monomial*, *binomial*, and *trinomial* when referring to these special polynomials and just call all the rest *polynomials*.

The **degree of a polynomial** and the degree of its terms are determined by the exponents of the variable. A monomial that has no variable, just a constant, is a special case. The **degree of a constant** is 0.

Definition: DEGREE OF A POLYNOMIAL

- The
*degree of a term*is the sum of the exponents of its variables. - The
*degree of a constant*is 0. - The
*degree of a polynomial*is the highest degree of all its terms.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms. Let's start by looking at a monomial. The monomial (8ab^2) has two variables (a) and (b). To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of (b) is 2. The sum of the exponents, 1+2,1+2, is 3 so the degree is 3.

Here are some additional examples.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in **standard form of a polynomial**. Get in the habit of writing the term with the highest degree first.

Example (PageIndex{1})

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

- (7y2−5y+3)
- (−2a^4b^2)
- (3x5−4x3−6x2+x−8)
- (2y−8xy^3)
- (15)

**Answer**Polynomial Number of terms Type Degree of terms Degree of polynomial ⓐ (7y^2−5y+3) 3 Trinomial 2, 1, 0 2 ⓑ (−2a^4b^2−2a^4b^2) 1 Monomial 4, 2 6 ⓒ (3x5−4x3−6x2+x−8) 5 Polynomial 5, 3, 2, 1, 0 5 ⓓ (2y−8xy^3) 2 Binomial 1, 4 4 ⓔ (15) 1 Monomial 0 0

Example (PageIndex{2})

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

- (−5)
- (8y^3−7y^2−y−3)
- (−3x^2y−5xy+9xy^3)
- (81m^2−4n^2)
- (−3x^6y^3z)

**Answer a**monomial, 0

**Answer b**polynomial, 3

**Answer c**trinomial, 3

**Answer d**binomial, 2

**Answer b**monomial, 10

Example (PageIndex{3})

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

- (64k^3−8)
- (9m^3+4m^2−2)
- (56)
- (8a^4−7a^3b−6a^2b^2−4ab^3+7b^4)
- (-p^4q^3)

**Answer**ⓐbinomial, 3 ⓑ trinomial, 3 ⓒ monomial, 0 ⓓ polynomial, 4 ⓔ monomial, 7

## Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.

Example (PageIndex{4})

Add or subtract:

- (25y^2+15y^2)
- (16pq^3−(−7pq^3)).

**Answer a**( egin{array} {ll} {} &{25y^2+15y^2} { ext{Combine like terms.}} &{40y^2} end{array} onumber )

**Answer b**( egin{array} {ll} {} &{16pq^3−(−7pq^3)} { ext{Combine like terms.}} &{23pq^3} end{array} onumber )

Example (PageIndex{5})

Add or subtract:

- (12q^2+9q^2)
- (8mn^3−(−5mn^3)).

**Answer**ⓐ (21q^2) ⓑ (13mn^3)

Example (PageIndex{6})

Add or subtract:

- (−15c^2+8c^2)
- (−15y^2z^3−(−5y^2z^3))

**Answer**ⓐ (−7c^2) ⓑ (−10y^2z^3)

Remember that like terms must have the same variables with the same exponents.

Example (PageIndex{7})

Simplify:

- (a^2+7b^2−6a^2)
- (u^2v+5u^2−3v^2)

**Answer**ⓐ Combine like terms.

(a^2+7b^2−6a^2 ;=; −5a^2+7b^2)

ⓑ There are no like terms to combine. In this case, the polynomial is unchanged.

(u^2v+5u^2−3v^2)

Example (PageIndex{8})

Add:

- (8y^2+3z^2−3y^2)
- (m^2n^2−8m^2+4n^2)

**Answer**ⓐ (5y^2+3z^2)

ⓑ (m^2n^2−8m^2+4n^2)

Example (PageIndex{9})

Add:

- (3m^2+n^2−7m^2)
- (pq^2−6p−5q^2)

**Answer**ⓐ (−4m^2+n^2)

ⓑ (pq^2−6p−5q^2)

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example (PageIndex{10})

Find the sum: ((7y^2−2y+9);+;(4y^2−8y−7)).

**Answer**(egin{align*} & ext{Identify like terms.} & & (underline{underline{7y^2}}−underline{2y}+9)+(underline{underline{4y^2}}−underline{8y}−7) [6pt]

& ext{Rewrite without the parentheses,}

& ext{rearranging to get the like terms together.} & & underline{underline{7y^2+4y^2}}−underline{2y−8y}+9−7[6pt]

& ext{Combine like terms.} & & 11y^2−10y+2 end{align*} )

Example (PageIndex{11})

Find the sum: ( (7x^2−4x+5);+;(x^2−7x+3))

**Answer**(8x^2−11x+8)

Example (PageIndex{12})

Find the sum: ((14y^2+6y−4);+;(3y^2+8y+5))

**Answer**(17y^2+14y+1)

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

Example (PageIndex{13})

Find the difference: ((9w^2−7w+5);−;(2w^2−4))

**Answer**(egin{align*} & & & (9w^2−7w+5);−;(2w^2−4) [6pt]

& ext{Distribute and identify like terms.} & & underline{underline{9w^2}}−underline{7w}+5-underline{underline{2w^2}}+4 [6pt]

& ext{Rearrange the terms.} & & underline{underline{9w^2-2w^2}}−underline{7w}+5+4[6pt]

& ext{Combine like terms.} & & 7w^2−7w+9 end{align*} )

Example (PageIndex{14})

Find the difference: ((8x^2+3x−19);−;(7x^2−14))

**Answer**(x^2+3x−5)

Example (PageIndex{15})

Find the difference: ((9b^2−5b−4);−;(3b^2−5b−7))

**Answer**(6b^2+3)

Example (PageIndex{16})

Subtract ((p^2+10pq−2q^2)) from ((p^2+q^2)).

**Answer**(egin{align*} & & & (p^2+q^2);−;(p^2+10pq−2q^2) [6pt]

& ext{Distribute and identify like terms.} & & underline{underline{p^2}}+underline{q^2}-underline{underline{p^2}}-10pq + underline{2q^2} [6pt]

& ext{Rearrange the terms, putting like terms together.} & & underline{underline{p^2-p^2}}−10pq +underline{q^2 + 2q^2}[6pt]

& ext{Combine like terms.} & & −10pq+3q^2 end{align*} )

Example (PageIndex{17})

Subtract ((a^2+5ab−6b^2)) from ((a^2+b^2))

**Answer**(−5ab+7b^2)

Example (PageIndex{18})

Subtract ((m^2−7mn−3n^2)) from ((m^2+n^2)).

**Answer**7mn+4n^2

Example (PageIndex{19})

Find the sum: ((u^2−6uv+5v^2);+;(3u^2+2uv))

**Answer**(egin{align*} & & & (u^2−6uv+5v^2);+;(3u^2+2uv) [6pt]

& ext{Distribute and identify like terms.} & & underline{underline{u^2}}-underline{6uv}+5v^2+underline{underline{3u^2}}+ underline{2uv} [6pt]

& ext{Rearrange the terms to put like terms together.} & & underline{underline{u^2}}+underline{underline{3u^2}}- underline{6uv}+ underline{2uv}+5v^2[6pt]

& ext{Combine like terms.} & & 4u^2−4uv+5v^2 end{align*} )

Example (PageIndex{20})

Find the sum: ((3x^2−4xy+5y^2);+;(2x^2−xy))

**Answer**(5x^2−5xy+5y^2)

Example (PageIndex{21})

Find the sum: ((2x^2−3xy−2y^2);+;(5x^2−3xy))

**Answer**(7x^2−6xy−2y^2)

When we add and subtract more than two polynomials, the process is the same.

Example (PageIndex{22})

Simplify: ((a^3−a^2b);−;(ab^2+b^3);+;(a^2b+ab^2))

**Answer**(egin{align*} & & & (a^3−a^2b);−;(ab^2+b^3);+;(a^2b+ab^2) [6pt]

& ext{Distribute} & & a^3−a^2b − ab^2 - b^3 + a^2b+ab^2[6pt]

& ext{Rearrange the terms to put like terms together.} & & a^3−a^2b + a^2b− ab^2 + ab^2 - b^3 [6pt]

& ext{Combine like terms.} & & a^3−b^3 end{align*} )

Example (PageIndex{23})

Simplify: ((x^3−x^2y);−;(xy^2+y^3);+;(x^2y+xy^2))

**Answer**(x^3+y^3)

Example (PageIndex{24})

Simplify: ((p^3−p^2q);+;(pq^2+q^3);−;(p^2q+pq^2))

**Answer**(p^3−3p^2q+q^3)

## Evaluate a Polynomial Function for a Given Value

A **polynomial function** is a function defined by a polynomial. For example, (f(x)=x^2+5x+6) and (g(x)=3x−4) are polynomial functions, because (x^2+5x+6) and (3x−4) are polynomials.

Definition: POLYNOMIAL FUNCTION

A *polynomial function* is a function whose range values are defined by a polynomial.

In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of (f(x)) for a given value of (x). To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

Example (PageIndex{26})

For the function (f(x)=3x^2+2x−15), find

- (f(3))
- (f(−5))
- (f(0)).

**Answer**ⓐ 18 ⓑ 50 ⓒ (−15)

Example (PageIndex{27})

For the function (g(x)=5x^2−x−4), find

- (g(−2))
- (g(−1))
- (g(0)).

**Answer**ⓐ 20 ⓑ 2 ⓒ (−4)

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

Example (PageIndex{28})

The polynomial function (h(t)=−16t^2+250) gives the height of a ball *t* seconds after it is dropped from a 250-foot tall building. Find the height after (t=2) seconds.

**Answer**( egin{array} {ll} {} &{h(t)=−16t^2+250} {} &{} { ext{To find }h(2) ext{, substitute }t=2.} &{h(2)=−16(2)^2+250} { ext{Simplify.}} &{h(2)=−16·4+250} {} &{} { ext{Simplify.}} &{h(2)=−64+250} {} &{} { ext{Simplify.}} &{h(2)=186} {} &{ ext{After 2 seconds the height of the ball is 186 feet.}} end{array} onumber )

Example (PageIndex{29})

The polynomial function (h(t)=−16t^2+150) gives the height of a stone *t* seconds after it is dropped from a 150-foot tall cliff. Find the height after (t=0) seconds (the initial height of the object).

**Answer**The height is (150) feet.

Example (PageIndex{30})

The polynomial function (h(t)=−16t^2+175) gives the height of a ball *t* seconds after it is dropped from a 175-foot tall bridge. Find the height after (t=3) seconds.

**Answer**The height is (31) feet.

## Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

Definition: ADDITION AND SUBTRACTION OF POLYNOMIAL FUNCTIONS

For functions (f(x)) and (g(x)),

[(f+g)(x)=f(x)+g(x)]

[(f−g)(x)=f(x)−g(x)]

Example (PageIndex{32})

For functions (f(x)=2x^2−4x+3) and (g(x)=x^2−2x−6), find: ⓐ ((f+g)(x)) ⓑ ((f+g)(3)) ⓒ ((f−g)(x)) ⓓ ((f−g)(−2)).

**Answer**ⓐ ((f+g)(x)=3x^2−6x−3)

ⓑ ((f+g)(3)=6)

ⓒ ((f−g)(x)=x^2−2x+9)

ⓓ ((f−g)(−2)=17)

Example (PageIndex{33})

For functions (f(x)=5x^2−4x−1) and (g(x)=x^2+3x+8), find ⓐ ((f+g)(x)) ⓑ ((f+g)(3)) ⓒ ((f−g)(x)) ⓓ ((f−g)(−2)).

**Answer**ⓐ ((f+g)(x)=6x^2−x+7)

ⓑ ((f+g)(3)=58)

ⓒ ((f−g)(x)=4x^2−7x−9)

ⓓ ((f−g)(−2)=21)

Access this online resource for additional instruction and practice with adding and subtracting polynomials.

__Adding and Subtracting Polynomials__

## Key Concepts

**Monomial**- A
**monomial**is an algebraic expression with one term. - A monomial in one variable is a term of the form axm,axm, where
*a*is a constant and*m*is a whole number.

- A
**Polynomials****Polynomial**—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.**monomial**—A polynomial with exactly one term is called a monomial.**binomial**— A polynomial with exactly two terms is called a binomial.**trinomial**—A polynomial with exactly three terms is called a trinomial.

**Degree of a Polynomial**- The
**degree of a term**is the sum of the exponents of its variables. - The
**degree of a constant**is 0. - The
**degree of a polynomial**is the highest degree of all its terms.

- The

## Glossary

**binomial**- A binomial is a polynomial with exactly two terms.

**degree of a constant**- The degree of any constant is 0.

**degree of a polynomial**- The degree of a polynomial is the highest degree of all its terms.

**degree of a term**- The degree of a term is the sum of the exponents of its variables.

**monomial**- A monomial is an algebraic expression with one term. A monomial in one variable is a term of the form axm,axm, where
*a*is a constant and*m*is a whole number.

**polynomial**- A monomial or two or more monomials combined by addition or subtraction is a polynomial.

**standard form of a polynomial**- A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.

**trinomial**- A trinomial is a polynomial with exactly three terms.

**polynomial function**- A polynomial function is a function whose range values are defined by a polynomial.

## 5.2: Add and Subtract Polynomials

Earl is building a doghouse whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house shown below, we can create an expression that combines several variable terms which allows us to solve this problem and others like it.

**Measurements of the front of the doghouse Earl is building.**

First, find the area of the square in square feet.

Then, find the area of the triangle in square feet.

Next, find the area of the rectangular door in square feet.

## POLYNOMIAL

The word polynomial is made by joining two words: “poly” - meaning many and “nomial” - meaning term, which means **"many terms".**

Let x be a variable, n be a positive integer and (a_0, a_1 , a_2. a_n) be real numbers (constants).

Then, (a_nX^n + a_

We use the notations f(x), g(x), h(x) etc. to denote a polynomial with the variable ‘x’.

Similarly, f(y), p(u) are polynomials with variables y and u respectively.

## Polynomials

Earl is building a doghouse, whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house, shown in [link], we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.

First find the area of the square in square feet.

Then find the area of the triangle in square feet.

Next find the area of the rectangular door in square feet.

The area of the front of the doghouse can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get 4 x 2 + 3 2 x − x ft 2 ,

In this section, we will examine expressions such as this one, which combine several variable terms.

### Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a **polynomial**, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as 384 π ,

is known as a **coefficient**. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product a i x i ,

is a **term of a polynomial**. If a term does not contain a variable, it is called a *constant*.

A polynomial containing only one term, such as 5 x 4 ,

is called a **monomial**. A polynomial containing two terms, such as 2 x − 9 ,

is called a **binomial**. A polynomial containing three terms, such as −3 x 2 + 8 x − 7 ,

is called a **trinomial**.

We can find the **degree** of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the **leading term** because it is usually written first. The coefficient of the leading term is called the **leading coefficient**. When a polynomial is written so that the powers are descending, we say that it is in standard form.

<div data-type="note" data-has-label="true" data-label="A General Note" markdown="1">

A **polynomial** is an expression that can be written in the form

Each real number *a _{i}*is called a

**coefficient**. The number a 0

that is not multiplied by a variable is called a *constant*. Each product a i x i

is a **term of a polynomial**. The highest power of the variable that occurs in the polynomial is called the **degree** of a polynomial. The **leading term** is the term with the highest power, and its coefficient is called the **leading coefficient**.

**Given a polynomial expression, identify the degree and leading coefficient**.

- Find the highest power of
*x*to determine the degree. - Identify the term containing the highest power of
*x*to find the leading term. - Identify the coefficient of the leading term.

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

- The highest power of
*x*is 3, so the degree is 3. The leading term is the term containing that degree, −4 x 3 .

The leading coefficient is the coefficient of that term,

The leading term is the term containing that degree,

The leading coefficient is the coefficient of that term,

The leading term is the term containing that degree,

The leading coefficient is the coefficient of that term,

Identify the degree, leading term, and leading coefficient of the polynomial 4 x 2 − x 6 + 2 x − 6.

The degree is 6, the leading term is − x 6 ,

and the leading coefficient is −1.

### Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, 5 x 2

are like terms, and can be added to get 3 x 2 ,

are not like terms, and therefore cannot be added.

**Given multiple polynomials, add or subtract them to simplify the expressions.**

( 12 x 2 + 9 x − 21 ) + ( 4 x 3 + 8 x 2 − 5 x + 20 )

We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.

( 2 x 3 + 5 x 2 − x + 1 ) + ( 2 x 2 − 3 x − 4 )

( 7 x 4 − x 2 + 6 x + 1 ) − ( 5 x 3 − 2 x 2 + 3 x + 2 )

Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.

( −7 x 3 − 7 x 2 + 6 x − 2 ) − ( 4 x 3 − 6 x 2 − x + 7 )

### Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

#### Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the 2

to obtain the equivalent expression 2 x + 14.

When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

**Given the multiplication of two polynomials, use the distributive property to simplify the expression.**

- Multiply each term of the first polynomial by each term of the second.
- Combine like terms.
- Simplify.

We can use a table to keep track of our work, as shown in [link]. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.

3 x 2 | − x | + 4 | |

2 x | 6 x 3 | −2 x 2 | 8 x |

+ 1 | 3 x 2 | − x | 4 |

#### Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the **f**irst terms, the **o**uter terms, the **i**nner terms, and then the **l**ast terms of each binomial.

The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.

**Given two binomials, use FOIL to simplify the expression.**

- Multiply the first terms of each binomial.
- Multiply the outer terms of the binomials.
- Multiply the inner terms of the binomials.
- Multiply the last terms of each binomial.
- Add the products.
- Combine like terms and simplify.

Use FOIL to find the product.

Find the product of the first terms.

Find the product of the outer terms.

Find the product of the inner terms.

Find the product of the last terms.

6 x 2 + 6 x − 54 x − 54 Add the products . 6 x 2 + ( 6 x − 54 x ) − 54 Combine like terms . 6 x 2 − 48 x − 54 Simplify .

Use FOIL to find the product.

#### Perfect Square Trinomials

Certain binomial products have special forms. When a binomial is squared, the result is called a **perfect square trinomial**. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.

Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.

When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.

**Given a binomial, square it using the formula for perfect square trinomials.**

- Square the first term of the binomial.
- Square the last term of the binomial.
- For the middle term of the trinomial, double the product of the two terms.
- Add and simplify.

Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.

#### Difference of Squares

Another special product is called the **difference of squares**, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let’s see what happens when we multiply ( x + 1 ) ( x − 1 )

The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let’s look at a few examples.

Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

**Is there a special form for the sum of squares?**

*No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.*

When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.

**Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.**

- Square the first term of the binomials.
- Square the last term of the binomials.
- Subtract the square of the last term from the square of the first term.

Multiply ( 9 x + 4 ) ( 9 x − 4 ) .

Square the first term to get ( 9 x ) 2 = 81 x 2 .

Square the last term to get 4 2 = 16.

Subtract the square of the last term from the square of the first term to find the product of 81 x 2 − 16.

Multiply ( 2 x + 7 ) ( 2 x − 7 ) .

### Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

Multiply ( x + 4 ) ( 3 x − 2 y + 5 ) .

Follow the same steps that we used to multiply polynomials containing only one variable.

Multiply ( 3 x − 1 ) ( 2 x + 7 y − 9 ) .

Access these online resources for additional instruction and practice with polynomials.

### Key Equations

perfect square trinomial | ( x + a ) 2 = ( x + a ) ( x + a ) = x 2 + 2 a x + a 2 |

difference of squares | ( a + b ) ( a − b ) = a 2 − b 2 |

### Key Concepts

- A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See [link].
- We can add and subtract polynomials by combining like terms. See [link] and [link].
- To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See [link].
- FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See [link].
- Perfect square trinomials and difference of squares are special products. See [link] and [link].
- Follow the same rules to work with polynomials containing several variables. See [link].

### Section Exercises

#### Verbal

Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.

You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

Use the distributive property, multiply, combine like terms, and simplify.

State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.

## How do you add two polynomials?

A polynomial is a sum of some powers of a certain variable, with some coefficient to multiply each power. Summing two polynomials simply means to sum the coefficients of the same powers, if this situations occour.

Let's say that your first polynomial is #1+x+2x^2-3x^3+15x^4# , and the second is #-3x^2+2x^3+5x^4-8x^5# . If we add them, the result is

#1+x+2x^2-3x^3+15x^4-3x^2+2x^3+5x^4-8x^5# . We can rearrange the terms so that the powers will be in order:

#1+x+2x^2-3x^2-3x^3+2x^3+15x^4+5x^4-8x^5#

At this point, you simply need to notice that:

- The constant term (i.e. #1# ) appears only in the first polynomial, so we have nothing to sum
- The same goes with the linear factor (i.e. #x# )
- The quadratic factor (i.e. #x^2# ) appears in both polynomial: in the first we have #2x^2# , in the second we have #-3x^2# . Summing the two coefficient, we have #2-3=-1# . The result is #-x^2#
- The cubic factor (i.e. #x^3# ) appears in both polynomial: in the first we have #-3x^3# , in the second we have #2x^3# . Summing the two coefficient, we have #-3+2=-1# . The result is #-x^3#
- The quartic factor (i.e. #x^4# ) appears in both polynomial: in the first we have #15x^4# , in the second we have #5x^4# . Summing the two coefficient, we have #15+5=20# . The result is #20x^4#
- The quintic factor (i.e. #x^5# ) appears only in the second polynomial, so we have nothing to sum

Finally, the answer is that the sum of the two polynomials is

#1+x-x^2-x^3+20x^4-8x^5#

## Polynomials

A polynomial is an expression made up of the sum of a finite number of powers in one or more variables multiplied by coefficients. This is a general polynomial in one variable:

Each coefficient ( (a_

Polynomials can have multiple variables, and get a little more complicated. Here is an example of a polynomial with two variables:

This lesson will focus on polynomials with only one variable raised to powers of two or less, these polynomials have the form:

The coefficients ( (a_<2>,a_<1>,a_<0>) ) can be positive, negative, or zero. If you know the coefficients of a polynomial you can fully construct it, because the ( x ) terms stay the same. For example say:

Can you write the polynomial associated with these coefficients? It would be:

This lesson will focus on adding, subtracting, multiplying, and factoring polynomials.

### Adding and subtraction polynomials

To add or subtract two polynomials you simply ‘combine like terms.’ This means that you add/subtract the coefficients of variables with the same power to get the new coefficients for those variables. For example, to add the polynomials ( 3x^<2>+6x-2 ) and ( x^<2>-4x+1 ) we would have:

Subtraction works in the same way. Subtracting ( 2x^<2>+x-3 ) from ( -x^<2>-4x+5 ) gives:

Take the same approach to add or subtract polynomials with two variables or more variables however, you will have more terms to keep track of.

### Multiplying polynomials

We are only going to focus on multiplying simple polynomials of the form ( a_<1>x+1_ <0>). To multiply two polynomials you must multiply each term in the first polynomial by each term in the second polynomial and vice versa. The lines in the following picture connect terms that we must multiply together.

The below example illustrates this multiplicative distribution. After the multiplication, we combine like terms to reach a simplified solution.

It is important to remember to multiply the ( x ) terms as well as the numbers. Below we show the solution to the general multiplication problem you can think of it as a template into which real numbers can be substituted for the ( a ), ( b ), ( c ), and ( d ) terms.

The key to multiplying polynomials is to make sure each term meets every other term once. Here is another example worth noting:

### Factoring polynomials

Factoring a polynomial means decomposing it into the product of two smaller polynomials. For example:

Essentially we are working in the opposite direction as we were above when we multiplied small polynomials. The small polynomials ( x-4 ) and ( x+3 ) are said to be factors of the larger polynomial ( x^<2>-x-12 ).

Finding the factors of a polynomial requires a bit of guesswork and a familiarity with the process of polynomial multiplication that we discussed in the last section. I’ll refer to the general equation below while describing the process of factoring polynomials:

When given an equation ( ax^<2>+bx+c ) and asked to factor it, you first need to choose ( A ) and ( C ) so that ( Acdot C=a ). Then you need to choose ( B ) and ( D ) such that ( Bcdot D=c ) and ( (Acdot D+Bcdot C)=b ).

As an example, say we want to factor the polynomial ( 2x^<2>+7x+3 ). We start by letting ( A=2 ) and ( C=1 ), since ( 2xcdot1x=2x^ <2>). Now we must find ( B ) and ( D ) such that ( Bcdot D=3 ) and ( (2D+B)=7 ). This step usually involves some ‘guessing and checking.’ You should end up with ( B=1 ) and ( D=3 ). We now have all the pieces to factor our polynomial:

You can check this factorization by multiplying the two factors you should get the original polynomial as the answer. Factoring polynomials requires good intuition. It can get especially tricky when you have multiple possibilities for A and C, like when factoring ( 4x^<2>-9x+2 ). In this case we could have either ( A=4 ), ( C=1 ) or ( A=2 ), ( C=2 ). What you need to do is just pick one of the two options and then try to find ( B ) and ( D ). If you’re unable to find a ( B ) and ( D ) that work you need to switch to the other combination of ( A ) and ( C ). In this case the correct factorization is:

## What is meant by Adding Polynomials?

Adding Polynomials is a matter of combining like terms with some order of operations considerations. To add polynomials add the coefficients of variables having the same power.

### How to add polynomials?

In general, the polynomial is written in descending order and the terms are written from the highest to the lowest degree. In Polynomial addition only like terms can be added. Like terms are the terms with the same variables raised to the same exponents. While adding only the co-efficient part changes and the variable part stays the same.

Follow the simple steps to add polynomials and get idea on how to do them manually.

If you are n't aware of the concept of the like terms you can add polynomials using column method i.e. line the polynomials up column-wise and add. Using the columns method can be great for complicated sums.

Seek help from free online calculators provided for multiple concepts of maths you might be looking for at onlinecalculator.guru

### FAQs on Adding Polynomials

**1. What is a Polynomial?**

Polynomials are expressions that are composed of many terms. The term “poly” means many and “nomial” means terms.

**2. How to add polynomials?**

To add polynomials simply identify the like terms and then add the terms.

**3. What are the different methods of adding polynomials?**

Different methods of adding polynomials are like term method and column method.

**4. Where do I get an elaborate solution to find addition of polynomials?**

You can get an elaborate solution to find the addition of polynomials on our page.

Now we'll look at an application of this skill and use the vertical method to solve.

Great Job! You should now be ready for subtracting polynomials.

### Comments

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## 5.2: Add and Subtract Polynomials

OPERATIONS WITH POLYNOMIALS

Adding and subtracting polynomials is simply the adding and subtracting of their like terms. There is a great similarity between the operations with polynomials and denominate numbers. Compare the following examples:

1. Add 5 qt and 1 pt to 3 qt and 2 pt.

One method of adding polynomials (shown in the above examples) is to place like terms in columns and to find the algebraic sum of the like terms. For example, to add 3a + b - 3c, 3b + c - d, and 2a + 4d, we would arrange the polynomials as follows:

Subtraction may be performed by using the same arrangement-that is, by placing terms of the subtrahend under the like terms of the minuend and carrying out the subtraction with due regard for sign. Remember, in subtraction the signs of all the terms of the subtrahend must first be mentally changed and then the process completed as in addition. For example, subtract 10a + b from 8a - 2b, as follows:

Again, note the similarity between this type of subtraction and the subtraction of denominate numbers.

Addition and subtraction of polynomials also can be indicated with the aid of symbols of grouping. The rule regarding changes of sign when removing parentheses preceded by a minus sign automatically takes care of subtraction.

For example, to subtract 10a + b from 8a - 2b, we can use the following arrangement:

Similarly, to add -3x + 2y to -4x - 5y, we can write

Practice problems. Add as indicated, in each of the following problems:

In problems 5 through 8, perform the indicated operations and combine like terms.

5. (2a + b) - (3a + 5b)

6. (5x3y + 3x2y) - (x3y)

7. (x + 6) + (3x + 7)

8. (4a 2 - b) - (2a 2 + b)

MULTIPLICATION OF A POLYNOMIAL BY A MONOMIAL

We can explain the multiplication of a polynomial by a monomial by using an arithmetic example. Let it be required to multiply the binomial expression, 7 - 2, by 4. We may write this 4 x (7 - 2)orsimply 4(7 - 2). Now 7 - 2 = 5. Therefore, 4(7 - 2) = 4(5) = 20. Now, let us then subtract. Thus, 4(7 - 2) = (4 x 7) - (4 x 2) = 20. Both methods give the same result. The second method makes use of the distributive law of multiplication.

When there are literal parts in the expression to be multiplied, the first method cannot be used and the distributive method must be employed. This is illustrated in the following examples:

Thus, to multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial.

## Polynomial class [closed]

**Want to improve this question?** Update the question so it's on-topic for Code Review Stack Exchange.

I'm looking for some assistance on an exercise for my C++ programming class. Unfortunately, I was rather ill the previous week an was unable to attend class, meaning I have only been able to use the textbook as a resource. I have done my best to complete the exercises, but, both because of my absence in class and my beginner status, I feel the code is rather flawed. I would really appreciate any corrections and suggestions, even moreso if you can elaborate on them.

Here are the instructions we were given for the assignment:

- Default constructor that dynamically allocates an array of DEFAULTPOLY elements (set to 50) and constructs a polynomial value of 0.
- A specialized (or alternate) constructor that takes an argument which indicates the size of the desired dynamic array and constructs a zero polynomial
- A copy constructor
- A destructor
- An assignment operator
- maxSize , which tells us the size of the currently allocated dynamic array
- grow , to allocate a new, larger dynamic array and fill it with the existing data
- setCoeff , to set a specific coefficient in the polynomial
- `retrieveCoeff* to get a specific coefficient from the polynomial
- *incrementCoeff`, to add a value to a specific coefficient in the polynomial
- degree , which determines the degree of the polynomial
- numOfTerms , which determines the number of terms in the polynomial (i.e., how many array elements are nonzero)
- evaluate , which evaluates the polynomial for a given value of X
- add , which adds one polynomial to another, changing the polynomial added to, growing the dynamic array if necessary
- subtract , which subtracts one polynomial from another, changing the polynomial subtracted from, growing the dynamic array if necessary
- derivative , which computes the derivative of a polynomial
- equals , which determines the equality of two polynomials
- `negate, which negates a polynomial
- multByConst , which multiplies a polynomial by a constant value

- A toString() function will be provided to you so that >all our polynomials will be displayed identically.
- The insertion operator is defined so we can easily print a polynomial
- The equality, inequality, and addition operators are provided and are simply defined in terms of your equals and add functions. You should not change any of the provided functions.

Below is the the .h file we were provided with:

An finally, here is the .cpp file I wrote:

Please let me know what changes and improvements I can make. Additionally, if you have any suggestions for the testing of the Poly class, those would also be welcomed.